## In-plane rotation classification for coherent X-ray imaging of single biomolecules |

Optics Express, Vol. 19, Issue 12, pp. 11691-11704 (2011)

http://dx.doi.org/10.1364/OE.19.011691

Acrobat PDF (3105 KB)

### Abstract

We report a new classification scheme with computation complexity well within the capacity of a PC for coherent X-ray imaging of single biomolecules. In contrast to current methods, which are based on data from large scattering angles, we propose to classify the orientations of the biomolecule using data from small angle scattering, where the signals are relatively strong. Further we integrate data to form radial and azimuthal distributions of the scattering pattern to reduce the variance caused by the shot noise. Classification based on these two distributions are shown to successfully recognize not only the patterns from molecules of the same orientation but also those that differ by an in-plane rotation.

© 2011 OSA

## 1. Introduction

1. D. Sayre, H. N. Chapman, and J. Miao, “On the extendibility of x-ray crystallography to noncrystals,” Acta Crystallogr., Sect. A: Found. Crystallogr. **54**, 323–239 (1998). [CrossRef]

4. S. Marchesini, H. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, “Coherent x-ray diffractive imaging: applications and limitations,” Opt. Express **11**, 2344–2353 (2003). [CrossRef] [PubMed]

5. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A **15**, 1662–1669 (1998). [CrossRef]

18. C. Song, H. Jiang, A. Mancuso, B. Amirbekian, L. Peng, R. Sun, S. S. Shah, Z. H. Zhou, T. Ishikawa, and J. Miao, “Quantitative imaging of single, unstained viruses with coherent x rays,” Phys. Rev. Lett. **101**, 158101 (2008). [CrossRef] [PubMed]

19. M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, and J.-M. Claverie, “Single mimivirus particles intercepted and imaged with an x-ray laser,” Nature **470**, 78–81 (2011). [CrossRef] [PubMed]

12. G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. **144**, 219–227 (2003). [CrossRef] [PubMed]

20. G. Bortel and G. Faigel, “Classification of continuous diffraction patterns: a numerical study,” J. Struct. Biol. **158**, 10–18 (2007). [CrossRef]

21. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phy. Rev. E **80**, 026705 (2009). [CrossRef]

22. R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. **5**, 64–67 (2009). [CrossRef]

23. D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules obtained from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys.: Condens. Matter **21**, 134014 (2009). [CrossRef]

## 2. Weak elastic scattering process and the issue of shot noise

*I*(

*x, y*), can be written as where

*r*is the free electron scattering length, Φ

_{e}*is the incident fluence and Ω*

_{inc}*is the solid angle of one detector pixel extended with respect to the biomolecule. The coordinates and atomic form factor of the*

_{pix}*n*atom in the biomolecule are described by (

^{th}*f*, respectively. The pixel coordinates for the detector array are (

_{n}*x,y,z*) and the wavelength of the X-ray is

*lambda*. The center region of the detector array has a hole to pass the unscattered photons. In this paper we are interested in the small-angle scattering patterns, the variation of the atomic form factor over the scattering angles can thus be ignored and

*f*is assumed to be a real function. The DNA polymerase delta (PDB ID:3IAY, molecular weight: 113822; total number of atoms: ∼ 8.2K) is used as an example of a typical biomolecule to calculate the diffraction pattern and perform orientation classifications.

_{n}^{14}and 2 × 10

^{12}per pulse. The beam is assumed to be 100 nm and the X-ray wavelength is 1.5

*Å*. The detector array is 150 mm away from the sample and the pixel size is 0.11 mm.

^{14}photons per pulse, the contrast of the speckles is diminished and some even disappear. For the 2 × 10

^{12}photons per pulse case, which is the current available level at LCLS, the number of photons scattered with scattering angles larger than 2° is so little that basically all speckles disappear. It will be very difficult to distinguish any exposures from each other, not to mention conducting orientation classification with this incident photon level.

*ρ*≤

_{min}*ρ*≤

*ρ*. We divide this area into regions of radial or azimuthal segments as shown in Fig. 2(a) and 2(b). The boundaries between these radial segments are rings with radii

_{max}*ρ*

_{1},

*ρ*

_{2},...,

*ρ*, which can be defined as where

_{N}*ρ*

_{0}= 3.72

*mm,*Δ

*ρ*

_{0}= 0.66

*mm*. We see that the width of the outer ring is larger than the inner rings. This will help to compensate for the decreasing signal strength with increasing radius. The radial distribution,

*I*(

_{ρ}*n*), can be written as, Note that

*I*is normalized with respect to the total scattered photons in the effective detection area. In this paper we choose

_{ρ}*N*= 60, which is able to significantly reduce the variance in the radial distribution and still give us sufficient sensitivity to rotations as we will see in the following parts of this paper.

*δθ*= 1°. As the scattering pattern is symmetric about the origin, we limit the value of

*θ*to be within [0, 180°]. The azimuthal distribution can be written as:

^{14}and 2 × 10

^{12}respectively. A typical radial and azimuthal distributions and their mean values are shown in Fig. 3(a) and 3(b) respectively. We see that when the incident photon number is 2 × 10

^{14}, the noisy version of the radial and azimuthal distribution is similar to its mean value. However when the incident photon number is 2 × 10

^{12}, the noisy version of the radial and azimuthal distribution deviates substantially from their corresponding mean values, which will affect the reliability of the classification adversely.

*Å*) by removing all atoms greater than 90

*Å*from the centroid and showed that the same process can be easily applied to spherical molecules.

## 3. Euler angles and the possibility of in-plane rotation

24. M. van Heel, “Angular reconstruction: a *posteriori* assignment of projection directions for 3d reconstruction,” Ultramicroscopy **21**, 111–124 (1987). [CrossRef] [PubMed]

26. M. van Heel, “Single-particle electron cryo-microscopy: towards atomic resolution,” Q. Rev. Biophys. **33**, 307–369 (2000). [CrossRef]

*ϕ*about an arbitrary axis can be described by three consecutive rotations, i.e., where

*R*is the rotation matrix that depends on the rotation axis and rotation angle

*ϕ*. The first rotation is about the intrinsic z-axis, the second rotation is about the intrinsic y-axis and the last one is again a rotation about the intrinsic z-axis.

*α*,

*β*and

*γ*to [0,

*π*].

*γ*-rotation is the so-called in-plane rotation because the z-axis of the intrinsic coordinate and the z-axis of the laboratory coordinates are co-axial. If we can classify the in-plane rotation out of all possible rotations and know the angle of the in-plane rotation of a selected pattern with respect to a reference pattern, we can rotate that pattern to overlap with the reference pattern. This will give us an opportunity to increase the SNR by averaging over all the in-plane rotations. The extent of the SNR improvement depends on the number of in-plane rotations.

*α*,

*β*,

*γ*) and can be written as [27] Due to the symmetry of the scattering pattern, the effective ranges for Euler angles are within [0,

*π*], thus we can write the effective probability density function in terms of

*α*, cos

*β*,

*γ*as

*π*

^{2}with Euler rotations defined by

*α*,

*γ*and cos

*β*. With a total number

*N*of random rotations, the number density for

_{total}*α*and

*γ*is

*β*. Thus the mean number of in-plane rotations is expected to be: where

*n*is the number of classes for each Euler angles. Thus the SNR of averaging over all the in-plane rotations can be further improved by

## 4. In-plane rotation recognition through the radial distribution

## 5. Finding the in-plane rotation angles through the azimuthal distribution

*I*. This angle information is needed if we hope to average over in-plane rotations. In this section we describe a simple method to find this angle.

_{θ}*γ*with respect to the reference pattern, then these two distributions should be shifted with respect to each other. This can be clearly seen in the surface plot shown in Fig. 4(f). The amount of the shift should be the angle of the in-plane rotation of the biomolecule with respect to its original position.

## 6. Accuracy of the classification vs photon budget

*°*, 180

*°*}. Twenty random values for each

*α*, cos

*β*,

*γ*are generated independently. Thus a total number of orientations is 20 × 20 × 20 = 8000, ignoring the inconsequential fact that the ranges for those three random variables (

*α*, cos

*β*,

*γ*) are not all equal to each other. The diffraction patterns for the protein with those orientations are generated using Eq. (1). Then we select an arbitrary pattern as a reference frame and assign the rest of the patterns to be the test patterns. The corresponding radial distribution for the reference frame and the test patterns are computed according to Eq. (3). Then the distance is calculated using Eq. (9) to compare the reference and test frames.

*N*of patterns from randomly oriented biomolecules, the total number of pairwise distances,

_{total}*N*, will be

_{d}*N*× (

_{total}*N*– 1) if each pattern is compared to other patterns in the data set. Note that the comparison between any two patterns are performed twice since each one has the opportunity to be a reference pattern. We also assume that the number of random values for each Euler angle is

_{total}*N*and

_{true}*N*are the number of true and assigned in-plane rotations, respectively. This accuracy rate can provide a quantitative measure of the classification performance. For a certain classification scheme, it depends mainly on two factors, noise level and the size of the orientation classes. For patterns that were found to be in-plane rotations, their values for

_{classified}*α*and

*β*should be identical in the ideal case, but with noise, we say the classification is accurate if

*α*and

*β*differ by an amount smaller than the size of the orientation class. In this section we will test the class size from 1° to 5°. In Table 1 we list the accuracy of these simulation results for different incident photon numbers and different orientation class sizes.

21. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phy. Rev. E **80**, 026705 (2009). [CrossRef]

22. R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. **5**, 64–67 (2009). [CrossRef]

*N*/

_{inplane}*N*, which is only 0.25% in this case.

_{total}*R*

^{3}space may not have been fully simulated in the above example. Thus we perform another round of simulations where the Euler angles are limited to

*α*∈ (0, 10°), cos

*β*∈ (0.9, 1). As the radial distributions are invariant to in-plane rotations, we choose the limit on the in-plane rotation angle still to be (0, 180°). For each Euler angle we generate 10 random numbers resulting in 1000 patterns. With a much smaller range of Euler angles, the percentage for two orientations whose Euler angles (

*α*,

_{i}*β*,

_{i}*i*= 1, 2) are within 1° becomes much higher. In our simulation, the percentage is about 48%, i.e., almost half of the orientations have a similar orientation. The accuracy in this small data set can be lower compared to the entire data set. In Table 1 we list the accuracy (numbers inside the parenthesis) for different photon numbers and orientation class sizes.

## 7. Discussion and summary

*α*,

*β*). With the proposed classification scheme we can achieve 79% accuracy even with current incident photon number. With a protein that is bigger than used in this paper or possesses additional symmetries, a good classification may be achieved with an even lower incident photon level. We have performed two simulations to test the performance of the proposed classification scheme. The performance for a real situation should be between these two cases.

22. R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. **5**, 64–67 (2009). [CrossRef]

*α*and

*β*. With fewer parameters to estimate, searching the manifold will be easier, which could potentially lower the required photon level even more. Further as the variance in the radial distributions is much smaller than those in the individual pixels, the searching of the 2-manifold will be more robust to noise, thus offering us a possibility to work with even fewer photons per pixel.

*a*, the maximum displacement of the same atom within one orientation class of size

*δθ*can be estimated as In order to maintain atomic resolution, the displacement of atoms within one class should be smaller than an angstrom. Thus for a biomolecule of size about 10

*nm*, such as the one considered in this paper, the size of the orientation class should be around 1.2°. From Table 1 we see that the classification scheme proposed in this paper can divide the data into orientation classes with size of 1°, thus maintaining the atomic resolution desired for coherent X-ray imaging.

28. J. Lipfert, D. Herschlag, and S. Doniach, “Riboswitch conformations revealed by small-angle x-ray scattering,” Methods Mol. Biol. **540**, 141–159 (2009). [CrossRef] [PubMed]

**5**, 64–67 (2009). [CrossRef]

24. M. van Heel, “Angular reconstruction: a *posteriori* assignment of projection directions for 3d reconstruction,” Ultramicroscopy **21**, 111–124 (1987). [CrossRef] [PubMed]

26. M. van Heel, “Single-particle electron cryo-microscopy: towards atomic resolution,” Q. Rev. Biophys. **33**, 307–369 (2000). [CrossRef]

21. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phy. Rev. E **80**, 026705 (2009). [CrossRef]

**5**, 64–67 (2009). [CrossRef]

**80**, 026705 (2009). [CrossRef]

**5**, 64–67 (2009). [CrossRef]

## Acknowledgment

## References and links

1. | D. Sayre, H. N. Chapman, and J. Miao, “On the extendibility of x-ray crystallography to noncrystals,” Acta Crystallogr., Sect. A: Found. Crystallogr. |

2. | J. Miao, Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature |

3. | R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature |

4. | S. Marchesini, H. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, “Coherent x-ray diffractive imaging: applications and limitations,” Opt. Express |

5. | J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A |

6. | I. Robinson, I. Vartanyants, G. Williams, M.A. Pfeifer, and J. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phy. Rev. Lett. |

7. | J. Miao, T. Ishkawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Ohodgson, “High resoluton 3D x-ray diffraction microscopy,” Phys. Rev. Lett. |

8. | J. C. H. Spence, U. Weierstall, and M. Howells, “Phase recovery and lensless imaging by iterative methods in optical x-ray and electron diffraction,” PPhilos. Trans. R. Soc. London, Ser. A |

9. | Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B |

10. | S. Marchesini, H. He, H. N. Chapman, S. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phy. Rev. B |

11. | G. Williams, M. A. Pfeifer, I. Vartanyants, and I. Robinson, “Three-dimensional imaging of microstructure in au nanocrystals,” Phys. Rev. Lett. |

12. | G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. |

13. | S. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, “Speden: reconstructing single particles from their diffraction patterns,” Acta Crystallogr. Sect. A: Found. Crystallogr. |

14. | D. Shapir, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. |

15. | H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A |

16. | H. N. Chapman, S. Bajt, A. Barty, W. H. Benner, M. J. Bogan, M. Frank, S. P. Hau-Riege, R. A. London, S. Marchesini, E. Spiller, A. Szke, and B. W. Woods, “Ultrafast coherent diffraction imaging with x-ray free-electron lasers,” Proc. FEL , 805–811 (2006). |

17. | J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard x-ray lensless imaging of extended objects,” Phy. Rev. Lett. |

18. | C. Song, H. Jiang, A. Mancuso, B. Amirbekian, L. Peng, R. Sun, S. S. Shah, Z. H. Zhou, T. Ishikawa, and J. Miao, “Quantitative imaging of single, unstained viruses with coherent x rays,” Phys. Rev. Lett. |

19. | M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, and J.-M. Claverie, “Single mimivirus particles intercepted and imaged with an x-ray laser,” Nature |

20. | G. Bortel and G. Faigel, “Classification of continuous diffraction patterns: a numerical study,” J. Struct. Biol. |

21. | N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phy. Rev. E |

22. | R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. |

23. | D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules obtained from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys.: Condens. Matter |

24. | M. van Heel, “Angular reconstruction: a |

25. | N. A. Farrow and F. P. Ottensmeyer, “A |

26. | M. van Heel, “Single-particle electron cryo-microscopy: towards atomic resolution,” Q. Rev. Biophys. |

27. | R. Miles, “On random rotations in |

28. | J. Lipfert, D. Herschlag, and S. Doniach, “Riboswitch conformations revealed by small-angle x-ray scattering,” Methods Mol. Biol. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(290.5840) Scattering : Scattering, molecules

(290.2558) Scattering : Forward scattering

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: February 15, 2011

Revised Manuscript: May 7, 2011

Manuscript Accepted: May 25, 2011

Published: June 1, 2011

**Virtual Issues**

Vol. 6, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Kaiqin Chu, James Evans, Nina Rohringer, Stefan Hau-Riege, Alexander Graf, Matthias Frank, Zachary J. Smith, and Stephen Lane, "In-plane rotation classification for coherent X-ray imaging of single biomolecules," Opt. Express **19**, 11691-11704 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11691

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### References

- D. Sayre, H. N. Chapman, and J. Miao, “On the extendibility of x-ray crystallography to noncrystals,” Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 323–239 (1998). [CrossRef]
- J. Miao, Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). [CrossRef]
- R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 753–757 (2000). [CrossRef]
- S. Marchesini, H. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, “Coherent x-ray diffractive imaging: applications and limitations,” Opt. Express 11, 2344–2353 (2003). [CrossRef] [PubMed]
- J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998). [CrossRef]
- I. Robinson, I. Vartanyants, G. Williams, M.A. Pfeifer, and J. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phy. Rev. Lett. 87(19), 195505 (2001). [CrossRef]
- J. Miao, T. Ishkawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Ohodgson, “High resoluton 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088302 (2002). [CrossRef]
- J. C. H. Spence, U. Weierstall, and M. Howells, “Phase recovery and lensless imaging by iterative methods in optical x-ray and electron diffraction,” PPhilos. Trans. R. Soc. London, Ser. A 360, 875–895 (2002). [CrossRef]
- Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003). [CrossRef]
- S. Marchesini, H. He, H. N. Chapman, S. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phy. Rev. B 68, 140101 (2003). [CrossRef]
- G. Williams, M. A. Pfeifer, I. Vartanyants, and I. Robinson, “Three-dimensional imaging of microstructure in au nanocrystals,” Phys. Rev. Lett. 90, 195505 (2003).
- G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003). [CrossRef] [PubMed]
- S. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, “Speden: reconstructing single particles from their diffraction patterns,” Acta Crystallogr. Sect. A: Found. Crystallogr. 60, 294–305 (2004). [CrossRef]
- D. Shapir, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005). [CrossRef]
- H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006). [CrossRef]
- H. N. Chapman, S. Bajt, A. Barty, W. H. Benner, M. J. Bogan, M. Frank, S. P. Hau-Riege, R. A. London, S. Marchesini, E. Spiller, A. Szke, and B. W. Woods, “Ultrafast coherent diffraction imaging with x-ray free-electron lasers,” Proc. FEL , 805–811 (2006).
- J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard x-ray lensless imaging of extended objects,” Phy. Rev. Lett. 98, 034801 (2007). [CrossRef]
- C. Song, H. Jiang, A. Mancuso, B. Amirbekian, L. Peng, R. Sun, S. S. Shah, Z. H. Zhou, T. Ishikawa, and J. Miao, “Quantitative imaging of single, unstained viruses with coherent x rays,” Phys. Rev. Lett. 101, 158101 (2008). [CrossRef] [PubMed]
- M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, and J.-M. Claverie, “Single mimivirus particles intercepted and imaged with an x-ray laser,” Nature 470, 78–81 (2011). [CrossRef] [PubMed]
- G. Bortel and G. Faigel, “Classification of continuous diffraction patterns: a numerical study,” J. Struct. Biol. 158, 10–18 (2007). [CrossRef]
- N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phy. Rev. E 80, 026705 (2009). [CrossRef]
- R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5, 64–67 (2009). [CrossRef]
- D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules obtained from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys.: Condens. Matter 21, 134014 (2009). [CrossRef]
- M. van Heel, “Angular reconstruction: a posteriori assignment of projection directions for 3d reconstruction,” Ultramicroscopy 21, 111–124 (1987). [CrossRef] [PubMed]
- N. A. Farrow and F. P. Ottensmeyer, “A posteriori determination of relative projection directions of arbitrarily oriented macromolecules,” J. Opt. Soc. Am. A 9, 1749–1760 (1992). [CrossRef]
- M. van Heel, “Single-particle electron cryo-microscopy: towards atomic resolution,” Q. Rev. Biophys. 33, 307–369 (2000). [CrossRef]
- R. Miles, “On random rotations in r3,” Biometrika 52, 636–639 (1965).
- J. Lipfert, D. Herschlag, and S. Doniach, “Riboswitch conformations revealed by small-angle x-ray scattering,” Methods Mol. Biol. 540, 141–159 (2009). [CrossRef] [PubMed]

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